Predictor variables sample variance

This GLS estimator is akin to inverse variance-weighted regression discussed in Section 8.2.3. Again there is a limitation V can be inverted only when the number of calibration samples is larger than the number of predictor variables, i.e. spectral wavelengths. Thus, one either has to work with a limited set of selected wavelengths or one must apply other solutions which have been proposed for tackling this problem [5]. 

Eor multivariate calibration in analytical chemistry, the partial least squares (PLS) method [19], is very efficient. Here, the relations between a set of predictors and a set (not just one) of response variables are modeled. In multicomponent calibration the known concentrations of / components in n calibration samples are collected to constitute the response matrix Y (n rows, / columns). Digitization of the spectra of calibration samples using p wavelengths yields the predictor matrix X (n rows, p columns). The relations between X and Y are modeled by latent variables for both data sets. These latent variables (PLS components) are constructed to exhaust maximal variance (information) within both data sets on the one hand and to be maximally correlated for the purpose of good prediction on the other hand. From the computational viewpoint, solutions are obtained by a simple iterative procedure. Having established the model for calibration samples. comp>o-nent concentrations for future mixtures can be predicted from their spectra. A survey of multi-component regression is contained in [20],... 

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